1Cademy - Choosing the Appropriate Special Product Pattern for $$(9b - 2)(2b + 9)$$, $$(9p - 4)^2$$, $$(7y + 1)^2$$, and $$(4r

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Choosing the Appropriate Special Product Pattern for $(2x - 3)(2x + 3)$ , $(8x - 5)^2$ , $(6m + 7)^2$ , and $(5x - 6)(6x + 5)$

Example

Choosing the Appropriate Special Product Pattern for $(9b - 2)(2b + 9)$ , $(9p - 4)^2$ , $(7y + 1)^2$ , and $(4r - 3)(4r + 3)$

Choose the correct pattern—Binomial Squares, Product of Conjugates, or general FOIL—for each product, then compute the result.

ⓐ $(9b - 2)(2b + 9)$ : The two binomials do not share the same pair of first and last terms, so neither special product pattern applies. Use the FOIL method: $(9b - 2)(2b + 9) = 18b^2 + 81b - 4b - 18 = 18b^2 + 77b - 18$

ⓑ $(9p - 4)^2$ : A single binomial is squared, so use the Binomial Squares Pattern $(a - b)^2 = a^2 - 2ab + b^2$ , with $a = 9p$ and $b = 4$ : $(9p - 4)^2 = (9p)^2 - 2(9p)(4) + 4^2 = 81p^2 - 72p + 16$

ⓒ $(7y + 1)^2$ : Again, a single binomial is squared. Use $(a + b)^2 = a^2 + 2ab + b^2$ , with $a = 7y$ and $b = 1$ : $(7y + 1)^2 = (7y)^2 + 2(7y)(1) + 1^2 = 49y^2 + 14y + 1$

ⓓ $(4r - 3)(4r + 3)$ : The binomials share the same first term $4r$ and the same last term $3$ , with one using subtraction and the other addition; they are conjugates. Apply the Product of Conjugates Pattern $(a - b)(a + b) = a^2 - b^2$ , with $a = 4r$ and $b = 3$ : $(4r - 3)(4r + 3) = (4r)^2 - 3^2 = 16r^2 - 9$

Updated 2026-04-29

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OpenStax Intermediate Algebra

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